Time evolution of quasi-stationary states

In this paper we study the time evolution of prepared states in some quantum mechanical models, and discuss the probability of decay and the rate of energy dissipation and their dependence on the form of the interaction. First we consider solvable models with divergent matrix elements for the operatorH ², whereH is the Hamiltonian of the system. We study two specific examples, one with well-defined eigenvalues and the other with renormalizable interaction. The time development of the initial state in the latter case depends on the cut-off parameter. In the second part of the paper, we show the possibility of existence of decaying states with long lifetime, where the amplitude of the initial state decreases like a Bessel function. In the third part, we determine the time development of a prepared state in a simple many-boson problem. Finally we study the problem of penetration of a wave packet through two phase-equivalent potential barriers, and we conclude that from the scattering phase shifts alone, it is not possible to determine the lifetime or the mode of decay of an unstable particle uniquely.